Riesz wavelets and generalized multiresolution analyses

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1 Appl. Comput. Harmon. Anal. 14 (2003) Riesz wavelets and generalized multiresolution analyses Marcin Bownik Department of Mathematics, University of Michigan, 525 East University Ave., Ann Arbor, MI , USA Received 22 August 2001; revised 3 October 2002; accepted 26 March 2003 Communicated by Pierre G. Lemarié-Rieusset Abstract We investigate Riesz wavelets in the context of generalized multiresolution analysis (GMRA). In particular, we show that Zalik s class of Riesz wavelets obtained by an MRA is the same as the class of biorthogonal wavelets associated with an MRA Elsevier Science (USA). All rights reserved. Keywords: Riesz wavelet; Multiresolution analysis; Generalized multiresolution analysis; Shift invariant space; Dimension function 1. Introduction The goal of this note is, among other things, to clarify what it means for a Riesz wavelet to be generated by an MRA. There are at least two ways in which we can say that a Riesz wavelet is associated with an MRA. Probably the most natural definition is the following which has appeared in the number of papers [1,13,15,18]. We say that a Riesz wavelet ψ L 2 (R) (with respect to dilation factor 2) is associated with an MRA if (V j ) j Z given by V j = W i, where W i = span { 2 i/2 ψ ( 2 i x k ) : k Z } i<j is an MRA. Here, { W i := f L 2 (R): f = w j,w i W i }, i<j i<j where the above series converges unconditionally. address: marbow@umich.edu /03/$ see front matter 2003 Elsevier Science (USA). All rights reserved. doi: /s (03)

2 182 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) On the other hand, Zalik proposed the following definition. We say that a Riesz wavelet ψ is obtained by an MRA if there is an MRA (V j ) j Z such that ψ V 1. There are several results in the subject. Kim et al. [13] in one dimension and Larson et al. [15] in higher dimensions have shown that a Riesz wavelet is a biorthogonal wavelet if and only if it is associated with a GMRA. Kim et al. [13] have also given a characterization of Riesz wavelets which are associated with an MRA in terms of certain dimension function. They have also shown that Riesz wavelets associated with an MRA always have dual (biorthogonal) Riesz wavelets (also associated with an MRA). This result is a refinement of Wang s characterization of biorthogonal wavelets associated with an MRA [18]. On the other hand, Zalik [19] has initiated investigation of the class of Riesz wavelets obtained by an MRA by giving a characterization of this class. The natural question concerns the relation between these different notions of Riesz wavelets. It is clear that any wavelet which is associated with an MRA is also obtained by the same MRA. However, the converse turns out to be quite a delicate question which has not been addressed in the literature, yet. In this paper we will show that the converse is also true under a necessary restriction on the number of wavelets relative to the order of the MRA. In particular, the two notions of Riesz wavelets associated with an MRA and obtained by an MRA turn out to be equivalent. The proof of this result is a consequence of a theorem of de Boor, DeVore, and Ron on the intersection of a nonstationary MRA. Theorem 1.1 [6, Theorem 4.9]. Suppose (φ j ) j Z is a sequence of functions in L 2 (R n ),wherez Z satisfies inf j Z j =.Let(U j ) j Z be the corresponding nonstationary (not necessarily nested) MRA given by U j = span { T 2 j kφ j : k Z n}. Then Y = j Z U j is a linear subspace of L 2 (R n ) of dimension 1. The paper is organized as follows. In Section 2 we show some basic properties of the dimension function of shift invariant spaces. In the next section we generalize Theorem 1.1 and use it to show our main result, Theorem 3.7. Finally, in the last section we present examples illustrating the optimality of the main result. 2. Shift invariant spaces In this section we recall some facts about shift invariant (SI) spaces that will be used in the sequel. Define the dilation operator D C f(x)= det C 1/2 f(cx),wherec is an n n nonsingular matrix, and the translation operator T y f(x)= f(x y), y R n. We use the following definition of the Fourier transform ˆ f(ξ)= R n f(x)e 2πi x,ξ dx. Definition 2.1. Suppose that Γ is a lattice, i.e., Γ = P Z n,wherep is an n n nonsingular matrix. We say that a closed subspace W L 2 (R n ) is shift invariant (SI) with respect to the lattice Γ,iff W

3 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) implies T γ f W for all γ Γ. Given a (countable) family Φ L 2 (R n ) and the lattice Γ we define the SI system E Γ (Φ) and SI space S Γ (Φ) by E Γ (Φ) ={T γ ϕ: ϕ Φ, γ Γ }, S Γ (Φ) = spane Γ (Φ). (2.1) When Γ = Z n we often drop the superscript Γ, and we simply say that W is SI. The dimension function of a SI space W is a Γ -periodic function dim Γ W : Rn N {0, }, which measures the size of W over the fibers of R n /Γ. Here, Γ is the dual lattice, i.e., Γ = { η R n : η,γ Z for γ Γ }. (2.2) That is, if Γ = P Z n then Γ = (P ) 1 Z n. The precise definition of the dimension function of a space W in terms of the range function is given by Proposition 2.6. Alternatively, the dimension function of a SI space W can be introduced by Stone s theorem as the multiplicity function of the projection valued measure coming from the representation of the lattice Γ acting on W via translations, see [1 3,11]. However, for our purposes, the following proposition can serve as a definition of the dimension function, where the notion of range function occurs only implicitly. Proposition 2.2. Suppose Φ L 2 (R n ) is a (countable) family and Γ = P Z n is a lattice. Then the dimension function of W = S Γ (Φ) is given by dim Γ W (ξ) = dim span{( ˆϕ(ξ + η) ) l 2( Γ ) : ϕ Φ }. (2.3) η Γ Moreover, if the system E Γ (Φ) is a tight frame with constant 1 for the space W = S Γ (Φ) then dim Γ W (ξ) = det P 1 ˆϕ(ξ + η) 2. (2.4) ϕ Φ η Γ Proof. The proof of Proposition 2.2 in the case Γ = Z n can be found, for example, in [5, Proposition 3.1], [7, Proposition 1.5, Theorem 2.5(ii)], and [16]. The general case follows by a change of variables, see also Lemma 2.3. Indeed, suppose that E Γ (Φ) is a tight frame with constant 1 for W = S Γ (Φ), Γ = P Z n. Then D P E Γ (Φ) = E Zn (D P Φ) is a tight frame for D P W which is SI with respect to Z n. By (2.4) applied for Z n and Lemma 2.3, dim Γ (( W P ) 1 ) ξ = dim Z n D P W (ξ) = D P ϕ(ξ + η) 2 = det P 1 (( ˆϕ P ) 1 ) (ξ + η) 2 ϕ Φ η Z n ϕ Φ η Z n = det P 1 (( ˆϕ P ) 1 ) ξ + η 2, ϕ Φ η Γ which shows (2.4) in the general case. Lemma 2.3. Suppose W L 2 (R n ) is SI with respect to the lattice Γ.LetC be any n n nonsingular matrix. Then D C W is SI with respect to the lattice C 1 Γ and the following identity holds: dim C 1 Γ D C W (ξ) = (( ) dimγ W C 1 ) ξ for a.e. ξ. (2.5) Proof. Suppose Φ L 2 (R n ) generates W, i.e., S Γ (Φ) = W. Then D C Φ generates D C W, i.e., S C 1Γ (D C Φ) = D C W. By (2.3),

4 184 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) dim C 1 Γ D C W (ξ) = dim span{( D c ϕ(ξ + η) ) l 2(( C 1 Γ ) ) } η (C 1 Γ) : ϕ Φ = dim span {( ˆϕ (( C ) 1 )) (ξ + η) l 2( C Γ ) : ϕ Φ } η C Γ = dim span {( ˆϕ (( C ) 1 )) ξ + η l 2( Γ ) : ϕ Φ } η Γ (( = dim ) Γ W C 1 ) ξ, since (C 1 Γ) = C Γ. Lemma 2.4. Suppose W L 2 (R n ) is SI with respect to the lattice Γ.LetΓ Γ be any sublattice of Γ. Then W is SI with respect to the lattice Γ, and, moreover, dim Γ W (ξ) = dim Γ W (ξ + d) for a.e. ξ, (2.6) d (Γ ) /Γ where the sum runs over representatives of distinct cosets of (Γ ) /Γ. Proof. Let C be any n n matrix such that Γ = CΓ, Γ = P Z n. It is well known that the orders of the quotient groups Γ/Γ and (Γ ) /Γ are the same and equal to det C. LetΦ L 2 (R n ) be any family whose translates by Γ generate a tight frame E Γ (Φ) with constant 1 for W = S Γ (Φ). Then E Γ (Φ) = ( E Γ ) (T d Φ) = E Γ T d Φ, d Γ/Γ d Γ/Γ and consequently by (2.4), dim Γ W (ξ) = det(cp ) 1 = det P 1 ϕ Φ ϕ d Γ/Γ T d Φ η (Γ ) d (Γ ) /Γ η Γ ˆϕ(ξ + η) 2 ˆϕ(ξ + η + d) 2 = dim Γ W (ξ + d). d (Γ ) /Γ Lemma 2.5. Suppose U,V L 2 (R n ) are SI with respect to the lattice Γ, and dim Γ U (ξ), dimγ V (ξ) are finite for a.e. ξ.thenw = U + V is also SI with respect to Γ, and dim Γ W (ξ) = dimγ U (ξ) + dimγ V (ξ) dimγ U V (ξ) for a.e. ξ. (2.7) Proof. The proof follows immediately from the range function interpretation of the dimension function, see Proposition 2.6. Let T : L 2 (R n ) L 2 (T n,l 2 (Z n )),wheret n =[ 1/2, 1/2) n, be the isometric isomorphism given by T f : T n l 2( Z n), T f(ξ)= ( f(ξ ˆ + k) ). (2.8) k Z n We recall that a range function is a mapping J : T n { closed subspaces of l 2( Z n)}. We also need the following fundamental description of SI spaces, see [5, Result 1.5], or [7, Proposition 1.5]. For simplicity, we state Proposition 2.6 for SI spaces with respect to the standard lattice Z n.

5 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) Proposition 2.6. A closed subspace V L 2 (R n ) is SI (with respect to Z n ) if and only if V = { f L 2( R n) : T f(ξ) J(ξ)for a.e. ξ T n}, where J is a measurable range function. The correspondence between V and J is one-to-one under the convention that the range functions are identified if they are equal a.e. Furthermore, the dimension function of V is given by dim V (ξ) = dim Zn V (ξ) = dim J(ξ) for a.e. ξ Tn. 3. GMRAs In this section we recall already known results about Riesz wavelets and GMRAs, extend a result of de Boor, DeVore, and Ron mentioned in Section 1, and show our main result. We start by recalling the notion of a GMRA, which has been studied by a number of authors [1 4,9,15]. Definition 3.1. Let A be an n n integer expansive dilation matrix which is fixed throughout this section. A generalized multiresolution analysis (GMRA) is a sequence of closed subspaces (V j ) j Z satisfying: V j V j+1, D A V j = V j+1, V j = L 2( R n), V j ={0}, j Z j Z T k V 0 = V 0 for all k Z n. The space V 0 is often called a core space. Given a Riesz wavelet Ψ ={ψ 1,...,ψ L } we define the sequence (V j (Ψ )) j Z by V j = V j (Ψ ) = span { D A i T k ψ: i<j,k Z n,ψ Ψ }. (3.1) It is easy to see that the sequence (V j ) j Z satisfies all the properties of GMRA except possibly the last one, i.e., that the core space V 0 is SI. In the case when (V j (Ψ )) j Z forms a GMRA, we say that Ψ is associated with a GMRA, or Ψ generates a GMRA. It turns out that the core space V 0 is SI if and only if Ψ is a biorthogonal wavelet, i.e., there exists a Riesz wavelet Φ ={φ 1,...,φ L } such that DA j T k ψ l,d A j T k φl = δj,j δ k,k δ l,l for all j,j Z, k,k Z n,l,l = 1,...,L. This was shown by Kim et al. [13] in one dimension and by Larson et al. [15] in higher dimensions. Theorem 3.2. Let Ψ ={ψ 1,...,ψ L } be a Riesz wavelet. The following are equivalent: (i) the sequence (V j (Ψ )) j Z forms a GMRA, (ii) Ψ is a biorthogonal wavelet, (iii) there exists an orthonormal wavelet Φ ={φ 1,...,φ L } which is associated with the same GMRA as Ψ, i.e., V 0 (Φ) = V 0 (Ψ ).

6 186 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) Conversely, Baggett et al. [1] have characterized those GMRAs that can be generated by orthonormal wavelets, and thus by Riesz wavelets by Theorem 3.2. Theorem 3.3. Let (V j ) j Z be a GMRA. The following are equivalent: (i) there exists an orthonormal wavelet Ψ ={ψ 1,...,ψ L } which is associated with (V j ) j Z, (ii) there exists a Riesz wavelet Ψ ={ψ 1,...,ψ L } which is associated with (V j ) j Z, (iii) the dimension function dim V0 (ξ) of the core space V 0 is finite for a.e. ξ and it satisfies the consistency equation (( dim V0 A ) 1 ) (ξ + d) = dimv0 (ξ) + L, for a.e. ξ, (3.2) d D where D is the set of det A representatives of different cosets of Z n /A Z n. Following Zalik [19] we can introduce the notion of wavelets obtained by a GMRA. Definition 3.4. We say that a GMRA (V j ) j Z is admissible of order L, i.e., the core space V 0 satisfies the consistency equation (3.2). We say that a Riesz wavelet Ψ ={ψ 1,...,ψ L } is obtained by a GMRA if there is an admissible GMRA (V j ) j Z of order L such that ψ 1,...,ψ L V 1. Clearly, every Riesz wavelet which is associated with a GMRA is also obtained by the same GMRA. The converse to this, see Theorem 3.7, is much less obvious and requires hard work. We start by generalizing Theorem 1.1 on the intersection of a nonstationary MRA to the case of general dilations and higher multiplicities. Theorem 3.5. Suppose (Φ j ) j Z is a sequence of finite subsets of L 2 (R n ) of cardinality L,whereZ Z satisfies inf j Z j =.Let(U j ) j Z be the corresponding nonstationary (not necessarily nested) GMRA given by U j = S A j Z n (Φ j ) = span { T A j kφ: φ Φ j,k Z n}. (3.4) Then Y = j Z U j is a linear subspace of L 2 (R n ) of dimension L. To show Theorem 3.5 we need a lemma describing linear independence of a finite set of measurable functions in terms of their values on some dense subset of R n. Lemma 3.6. Suppose D is a countable and dense subset of R n. Suppose f 1,...,f m : R n C are any measurable functions. The following are equivalent: (i) f 1,...,f m are linearly dependent, i.e., there exist (c 1,...,c m ) C m \{0} such that m c l f l (x) = 0 for a.e. x R n, l=1 (3.3)

7 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) (ii) for almost every x R n there exist (c 1 (x),...,c m (x)) C m \{0} such that m c l (x)f l (x + d) = 0 for all d D. l=1 Proof. The implication (i) (ii) is trivial. Suppose (ii) holds. Define f : R n C m by f(x)= (f 1 (x),..., f m (x)). Following the terminology of Helson [5,7,10], we define the range function J : R n {subspaces of C m } by J(x)= span { f(x+ d): d D }. Let P(x) denote the orthogonal projection onto J(x). It is clear that J(x) is measurable, i.e., for any v,w C m the scalar function p v,w (x) = P(x)v,w is measurable. Clearly, p v,w (x) = p v,w (x + d) for all d D. (3.5) Let x,y R n be two Lebesgue points of the function p v,w,wherev,w C m. Recall that if g L 1 loc (Rn ) then a point x R n is said to be a Lebesgue point of g if lim B 0, x B 1 B B g(z)dz = g(x) with the limit taken over balls B. The Lebesgue Differentiation theorem asserts that almost every point x R n is a Lebesgue point of g L 1 loc (Rn ). Let (B i ) i N be a sequence of balls such that x B i and B i 0asi. By the density of D there exists a sequence (d i ) i N D such that y d i B i for all i N. By the Lebesgue Differentiation theorem and (3.5), 1 p v,w (x) = lim i B i B i 1 p v,w (z) dz = lim i B i B i p v,w (z + d i ) dz = p v,w (y). Hence, p v,w (x) = const for a.e. x. Since v,w C m are arbitrary J(x)= const for a.e. x. By (ii), J(x) C m for a.e. x, and hence there exists a hyperplane H C m such that J(x) H for a.e. x. Therefore, f(x) H for a.e. x, and (i) holds. Proof of Theorem 3.5. Let f 1,...,f m be arbitrary functions in Y,wherem = L + 1. It suffices to show that f 1,...,f m axe linearly dependent. Let j Z and let Φ j ={φ 1,...,φ L }. As a consequence of [5, Theorem 1.7] and a change of variables g U j if and only if L ĝ(ξ) = τ l (ξ) ˆφ l (ξ) for all ξ R n, l=1 for some measurable and (A ) j Z n -periodic functions τ 1,...,τ L, i.e., τ l (ξ) = τ l( ξ + ( A ) j ) k for all k Z n. Therefore, ˆ f k (ξ) = L τk l (ξ) ˆφ l (ξ) for k = 1,...,m, (3.6) l=1

8 188 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) for some measurable and (A ) j Z n -periodic functions τk l(ξ).let(c 1(ξ),..., c m (ξ)) be a nonzero solution to the undetermined system of linear equations c 1 (ξ)τ1 1(ξ) + +c m(ξ)τm 1 (ξ) = 0, (3.7) c 1 (ξ)τ1 L(ξ) + +c m(ξ)τm L(ξ) = 0. By (3.6) and (3.7) we have m k=1 c k (ξ) ˆ f k (ξ) = 0 and by the periodicity of τk l(ξ), m ( ( c k (ξ) fˆ k ξ + A ) j ) k = 0 forall k Z n. (3.8) k=1 Let D = j Z (A ) j Z n. Since inf j Z j =, D is dense in R n.sincea (and thus A ) preserves the lattice Z n we can find universal c 1 (ξ),..., c m (ξ) yielding (3.8) for all j Z. Therefore, m c k (ξ) fˆ k (ξ + d) = 0 forall d D. k=1 By Lemma 3.6, ˆ f 1,..., ˆ f m are linearly dependent which completes the proof of Theorem 3.5. We are now ready to state our main result. Theorem 3.7. Suppose Ψ ={ψ 1,...,ψ L } is a Riesz wavelet, and (V j ) j Z is an admissible GMRA of order L, i.e., dim V0 (ξ) < for a.e. ξ and (3.2) holds. The following are equivalent: (i) Ψ is obtained by the GMRA (V j ) j Z, i.e., (3.3) holds, (ii) W j + V j = V j+1 and W j V j ={0} for all j Z, (3.9) where W j := span { D A j T k ψ: k Z n,ψ Ψ }, (3.10) (iii) Ψ is associated with the GMRA (V j ) j Z, i.e., V j (Ψ ) := i<j W i = V j for all j Z. (3.11) The implications (iii) (ii) (i) are trivial. It remains to show the difficult implication (i) (iii). We will present two different proofs of this implication. The first proof works only for GMRAs of finite height, i.e., those for which the dimension function of the core space V 0 is essentially bounded, and uses Theorem 3.5. This is the most interesting situation, since it covers Riesz wavelets obtained by an MRA (also MRAs with higher than one multiplicities). However, the second proof works for general GMRAs of possibly infinite height.

9 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) Proof of Theorem 3.7 for finite height GMRAs. Suppose Ψ is obtained by the GMRA (V j ) j Z.For j Z let W j be given by (3.10). Since W j V j+1 for all j Z we have that W j V 0. (3.12) j<0 Here, j<0 W j consists of all functions f L 2 (R n ) which can be represented as f = j<0 w j,where the series converges unconditionally and w j W j. To complete the proof of Theorem 3.7 it remains to show that we have the equality in (3.12). Indeed, suppose that V 0 (Ψ ) := j<0 W j = V 0. By applying the dilation operator ( V j (Ψ ) = D A j V0 (Ψ ) ) = D A j (V 0 ) = V j for all j Z. Therefore, the sequence (V j (Ψ )) j Z is exactly the same as the GMRA (V j ) j Z, and hence Ψ is associated with the GMRA (V j ) j Z. To show the equality in (3.12), note that W j is SI with respect to the lattice A j Z n and its dimension function dim A j Z n W j (ξ) = L for a.e. ξ, (3.13) since D A j (E Zn (Ψ )) = E A j Z n (D A j Ψ) forms a Riesz basis for W j. Define a nonstationary GMRA (U j ) j 0 by ( U j = V 0 W i ). (3.14) j i<0 Since j i<0 W i is SI with respect to A j Z n, hence its orthogonal complement is also SI with respect to the same lattice, and therefore U j = V 0 ( j i<0 W i) is also SI with respect to A j Z n. We claim that for all j 0 the dimension function of U j coincides with the dimension function of V j, i.e., dim A j Z n U j (ξ) = dim A j Z n V j (ξ) for a.e. ξ. (3.15) Indeed, (3.15) holds for j = 0. By induction hypothesis suppose that (3.15) holds for some j 0. Then we need to show (3.15) for j 1. Since (W 1 + W 2 + +W j ) U j = V 0 = (W 1 + W 2 + +W j + W j 1 ) U j 1 = (W 1 + W 2 + +W j ) + (W j 1 U j 1 ), and (W 1 + W 2 + +W j ) U j ={0}=(W 1 + W 2 + +W j ) (W j 1 U j 1 ), we conclude by Lemma 2.5 that the dimension functions of U j and W j 1 U j 1 are the same, i.e., dim A j+1 Z n U j (ξ) = dim A j+1 Z n W j 1 U j 1 (ξ) = dim A j+1 Z n W j 1 (ξ) + dim A j+1 Z n U j 1 (ξ) = L + dim A j+1 Z n U j 1 (ξ). (3.16)

10 190 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) Observe that the consistency equation (3.2) can be written by Lemmas 2.3 and 2.4, (( dim Zn V 1 (ξ) = dim Zn D A (V 0 )(ξ) = dimazn V 0 A ) 1 ) ξ (( = dim ) Zn V 0 A 1 ) ξ + d = dim Z n V 0 (ξ) + L. d (A ) 1 Z n /Z n Hence, for j Z by the above and Lemma 2.3, (( dim A j+1 Z n V j A ) 1 ) (( ξ = dim A j+1 Z n D A j (V 0 ) A ) 1 ) (( ξ = dim AZ n V 0 A ) j 1 ) ξ (( = dim ) Zn V 0 A j ) ξ + L = dim A j Z n V j (ξ) + L. (3.17) On the other hand, by Lemma 2.4 and the induction hypothesis, dim A j+1 Z n A j Zn U j (ξ) = dimu j (ξ + d) d (A ) j 1 Z n /(A ) j Z n A j Zn = dimv j d (A ) j 1 Z n /(A ) j Z n Combining (3.16) (3.18), and Lemma 2.3, (ξ + d) = dim A j+1 Z n V j (ξ). (3.18) dim A j+1 Z n U j 1 (ξ) = dim A j+1 Z n U j (ξ) L = dim A j+1 Z n V j (ξ) L ( = dim A j Z n V j A ξ ) ( = dim A j Z n D A V j 1 A ξ ) = dim A j+1 Z n V j 1 (ξ), we obtain (3.15) for j 1. Therefore, by induction, (3.15) holds for all j 0. Suppose that our GMRA (V j ) j Z has a finite height L. Therefore, (( dim A j Z n V j (ξ) = dim ) Zn V 0 A j ) ξ L for a.e. ξ. By (3.15) this implies for all j 0, dim A j Z n U j (ξ) L. This means that for each j 0 we can find a generating set Φ j of cardinality at most L such that U j = S A j Z n (Φ j ), see [5, Theorem 3.5]. By Theorem 3.5, this implies that Y = j 0 U j is a linear subspace of L 2 (R n ) of dimension L. By (3.14), ( Y = V 0 W j ), (3.19) j<0 which represents a defect from the equality in (3.12). It remains to show that the defect space Y is trivial, i.e., Y ={0}. This can be most easily seen by considering an auxiliary defect space Ỹ defined by ( Ỹ = V 0 W j ). (3.20) j 0 In other words, Ỹ is the unique complementary space of j<0 W j inside V 0, which is, in addition, contained in j 0 W j, i.e., ( ) ( ) W j + Ỹ = V 0, W j Ỹ ={0}, and Ỹ W j. j 0 j<0 j<0

11 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) The defect space Ỹ must have the same dimension as Y. Moreover, since V 0 and j 0 W j are SI with respect to Z n, Ỹ is SI, too. Since the trivial space is the only finitely-dimensional SI space, the defect space Ỹ must be trivial, and consequently, W j = V 0. j<0 This completes the proof of Theorem 3.7 in the finite height case. Proof of Theorem 3.7 for infinite height GMRAs. The proof follows along the lines of the argument in the finite height case. As before, we could define a nonstationary GMRA (U j ) j 0, the defect space Y by (3.19), and the auxiliary SI defect space Ỹ by (3.20). The defect spaces Y and Ỹ have the same (possibly infinite) dimension. Even though, a priori, Y (and thus Ỹ ) may not be finite-dimensional, we will see that Ỹ has to be in a certain sense arbitrarily small, and thus trivial. Note first that if (V j ) j Z is an admissible GMRA of order L, T n dim Zn V 0 (ξ) dξ = L det A 1, (3.21) where T n is a fundamental domain, say T n =[ 1/2, 1/2) n. Indeed, (3.21) follows from the fact that for any orthonormal wavelet Ψ ={ ψ 1,..., ψ L } associated with (V j ) j Z,wehave dim Zn V 0 (ξ) = D Ψ (ξ) := ˆψ (( A ) j ) (ξ + k) 2, ψ Ψ j=1 k Z n see [7,8,12,15,17]. Since for any j<0, (W 1 + W 2 + +W j ) + Ỹ V 0 and (W 1 + W 2 + +W j ) Ỹ ={0}, therefore, dim A j Z n Ỹ 1 (ξ) + i=j dim A j Z n W i (ξ) dim A j Z n V 0 (ξ). By Lemma 2.4 and (3.13), dim A j Z n W i (ξ) = det A i j L, and thus dim A j Z n (ξ) dim A j Z n Ỹ V 0 (ξ) L det A j 1 det A 1. Therefore, integrating the above over (A ) j (T n ) we have dim Zn (ξ) dξ dim Zn Ỹ V 0 (ξ) dξ + L det A j 1 det A 1, (3.22) T n T n since for any SI space V (with respect to Z n )andj 0wehave dim A j Z n V (ξ) dξ = V (ξ + d)dξ = (A ) j (T n ) (A ) j (T n ) d (A ) j Z n /Z n dim Zn T n dim Zn V (ξ) dξ.

12 192 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) Therefore, by letting j in (3.22) we see that by (3.21), dim Zn (ξ) dξ = 0, Ỹ T n and consequently the defect space Ỹ is trivial. This completes the proof of Theorem 3.7 in the general case. As an immediate corollary of Theorems 3.2 and 3.7 we have. Corollary 3.8. Suppose a Riesz wavelet Ψ ={ψ 1,...,ψ L }, L = det A 1, is obtained by an MRA, i.e., there is an MRA (V j ) j Z such that ψ 1,...,ψ L V 1.ThenΨ is a biorthogonal wavelet associated with thesamemra(v j ) j Z. Remarks. (i) The assumption in Theorem 3.7 that (V j ) j Z is an admissible GMRA of order L can be relaxed a bit, but not too much. Indeed, it follows from the second proof of Theorem 3.7 that it suffices to assume (instead of admissibility) that a GMRA (V j ) j Z satisfies (3.21) or even less, T n dim Zn V 0 (ξ) dξ L det A 1. (3.23) In that case, any of the conditions (i) (iii) would imply that (V j ) j Z is an admissible GMRA of order L. Therefore, there are no Riesz wavelets Ψ that are obtained by a GMRA of a strictly lower order than the cardinality of Ψ. However, there are examples of Riesz wavelets Ψ that are obtained by a GMRA of a strictly higher order than the cardinality of Ψ, but which are not associated with a GMRA, see Example 4.4. Therefore, the assumption of admissibility in Theorem 3.7 is necessary. (ii) Corollary 3.8 can be combined with Zalik s characterization of wavelets obtained by an MRA [19, Proposition 2.2] to yield several interesting results about such wavelets, see [14]. 4. Examples One could think that the implication (ii) (iii) in Theorem 3.7 is immediate and follows from basic functional analysis. A slightly counterintuitive Example 4.1 shows that this is false. Example 4.1. Let (e i ) i Z be the standard orthonormal basis of l 2 (Z). Define the spaces (V i ) i Z and (W i ) i Z by { W i = span{e i }, V i = span...,e i 2,e i 1, c j e j }, j i where 0 <c<1. Clearly, (V i ) i Z satisfies GMRA like properties, i.e., V i V i+1, D(V i ) = V i+1, V i ={0}, V i = l 2 (Z), i Z i Z

13 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) where D is a dilation operator (known otherwise as the shift operator) acting on the standard basis by D(e i ) = e i+1. Moreover, V i + W i = V i+1, V i W i ={0}, (4.1) and the corresponding projections of V i+1 onto V i and W i are uniformly bounded for all i Z. Nevertheless, ( ) { V 0 = W i Y, where Y = span c j e j }, (4.2) i<0 and the defect space Y is nontrivial. Example 4.1 can be easily transplanted to the setting of SI spaces using Proposition 2.6. Example 4.2. Using the notation from Example 4.1, define the SI spaces Ṽ i, W i,andỹ by j 0 Ṽ i = { f L 2 (R): T f(ξ) V i for a.e. ξ [ 1/2, 1/2) }, W i = { f L 2 (R): T f(ξ) W i for a.e. ξ [ 1/2, 1/2) }, Ỹ = { f L 2 (R): T f(ξ) Y for a.e. ξ [ 1/2, 1/2) }, where T f is given by (2.8). Clearly, (Ṽ i ) i Z satisfies GMRA like properties, i.e., Ṽ i Ṽ i+1, D(Ṽ i ) Ṽ i+1, Ṽ i ={0}, Ṽ i = L 2 (R), i Z where D is a dilation operator mapping f(ξ)to ˆ f(ξ ˆ 1) in the Fourier domain. Hence, D is just, the modulation operator given by Df(x) = e 2πix f(x). It is then obvious that Ṽ i and W i satisfy the analogues of (4.1) and (4.2), where the defect space Ỹ is nontrivial. i Z Example 4.2 would be more satisfying if the dilation operator D were the usual dyadic dilation operator. However, the author is unaware whether such construction is possible. The following simple example shows that the assumption of the admissibility of a GMRA in Theorem 3.7 is necessary, although it can be relaxed a bit by (3.23). In particular, Example 4.2 shows that the restriction on the number of wavelets being equal to the order of a GMRA is unavoidable. Example 4.3. Let (V i ) i Z be an MRA associated to the Haar wavelet ψ = 1 [0,1/2) 1 [1/2,1), i.e., V i = { f L 2 (R): f [k2 i,(k+1)2 i ) is constant for all k Z }. Define the misnumbered GMRA (V i ) i Z by V i = V i+1. Clearly, (V i ) i Z is an admissible GMRA of order 2 in the sense of Definition 3.4. In fact, (V i ) i Z is an MRA of order 2, with 1 [0,1/2) and 1 [1/2,1) being two scaling functions of V 0.Thenψ is obtained by (V i ) i Z, in the sense that ψ V 1, but it cannot be associated with (V i ) i Z, since by Theorem 3.3, L would have to be 2. Naturally, in this example, the Haar wavelet ψ can be replaced by any orthonormal dyadic wavelet in L 2 (R). A more elaborated Example 4.4 gives an example of a Riesz wavelet ψ which cannot be obtained by any admissible GMRA of order 1, but yet, it is obtained by some MRA of higher order.

14 194 M. Bownik / Appl. Comput. Harmon. Anal. 14 (2003) Example 4.4. Let ψ be a discontinuous perturbation of the Haar function, ψ = 1 [0,1/2 ε) 1 [1/2+ε,1), where 0 <ε<1/2. Zalik [19, Example 3.2] has shown that for sufficiently small ε>0 (say ε = 1/32), ψ is a Riesz wavelet which is not obtained by an MRA. Let (V i ) i Z be the classical Haar MRA as in Example 4.3. Define the GMRA (V i ) i Z, byv i = V i+4 for i Z. Clearly, (V i ) i Z is an admissible GMRA of order 16; more precisely, an MRA of order 16. Moreover, the perturbed Haar wavelet ψ with ε = 1/32 is obtained by the MRA (V i ) i Z (of order 16), that is, ψ V 1. However, ψ cannot be obtained by any admissible GMRA of order 1. On the contrary, suppose otherwise. By Theorem 3.7, ψ must be associated with an admissible GMRA (V i ) i Z order 1. Moreover, dim V (ξ) 1 for a.e. ξ, since ˆψ(ξ) 0 for a.e. ξ. On the other hand, dim V 0 (ξ) dξ = 1, hence dim V (ξ) = 1 for a.e. ξ, which means that (V 0 i ) i Z is an MRA. This is a contradiction with a fact due to Zalik that ψ is not obtained by an MRA. References [1] L.W. Baggett, H.A. Medina, K.D. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in R n, J. Fourier Anal. Appl. 5 (1999) [2] L.W. Baggett, K.D. Merrill, Abstract harmonic analysis and wavelets in R n, in: The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, TX, 1999), American Mathematical Society, Providence, RI, 1999, pp [3] L.W. Baggett, An abstract interpretation of the wavelet dimension function using group representations, J. Funct. Anal. 173 (2000) [4] J.J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998) [5] C. de Boor, R.A. DeVore, A. Ron, The structure of finitely generated shift-invariant spaces in L 2 (R d ), J. Funct. Anal. 119 (1994) [6] C. de Boor, R.A. DeVore, A. Ron, On the construction of multivariate (pre)wavelets, Constr. Approx. 9 (1993) [7] M. Bownik, The structure of shift invariant subspaces of L 2 (R n ), J. Funct. Anal. 177 (2000) [8] M. Bownik, Z. Rzeszotnik, D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001) [9] D. Han, D.R. Larson, M. Papadakis, Th. Stavropoulos, Multiresolution analyses of abstract Hilbert spaces and wandering subspaces, in: The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, TX, 1999), American Mathematical Society, Providence, RI, 1999, pp [10] H. Helson, Lectures on Invariant Subspaces, Academic Press, New York/London, [11] H. Helson, The Spectral Theorem, Springer, Berlin, [12] E. Hernández, G. Weiss, A First Course on Wavelets, in: Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, [13] H.O. Kim, R.Y. Kim, J.K. Lim, Characterizations of biorthogonal wavelets which are associated with biorthogonal multiresolution analyses, Appl. Comput. Harmon. Anal. 11 (2001) [14] H.O. Kim, R.Y. Kim, Y.H. Lee, J.K. Lim, On Riesz wavelets associated with multiresolution analyses, Appl. Comput. Harmon. Anal. 13 (2002) [15] D. Larson, W.-S. Tang, E. Weber, Riesz wavelets and multiresolution structures, in: Proc. SPIE, Vol. 4478, 2001, pp [16] A. Ron, Z. Shen, Frames and stable bases for shift-invariant subspaces of L 2 (R d ), Canad. J. Math. 47 (1995) [17] A. Ron, Z. Shen, The wavelet dimension function is the trace function of a shift-invariant system, Proc. Amer. Math. Soc. 131 (2003) [18] X. Wang, The study of wavelets from the properties of their Fourier transforms, Ph.D. thesis, Washington University in St. Louis, [19] R.A. Zalik, Riesz bases and multiresolution analyses, Appl. Comput. Harmon. Anal. 7 (1999)